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Appl. Comput. Math., V.9, N.1, 2010, pp.
QUASILINEARIZATION APPROACH FOR SOLVING VOLTERRA’S
K. PARAND1, M. GHASEMI2, S. REZAZADEH2, A. PEIRAVI2, A. GHORBANPOUR2,
A. TAVAKOLI GOLPAYGANI3
Abstract. The study used the Quasilinearization method to solve Volterra’s model for popu-
lation growth of a species within a closed system is proposed. This model is a nonlinear integro-
diﬀerential where the integral term represents the eﬀect of toxin. First we convert this model
to a nonlinear ordinary diﬀerential equation, then approximate the solution of this equation by
treating the nonlinear terms as a perturbation about the linear ones. Finally we compare this
method with the other methods and come to the conclusion that the Quasilinearization method
gives excellent results.
Keywords: Volterra’s Population Model, Quasilinearization, Integro-diﬀerential, Barycentric.
AMS Subject Classiﬁcation:
The Quasilinearization method was developed many years ago in the theory of linear pro-
gramming by Bellman and Kalaba [1, 5] as a generalization of the Newton Raphson method
[4, 11] to solve the systems of nonlinear ordinary and partial diﬀerential equations. Modern
developments and applications of the QLM method to diﬀerent ﬁelds are given in a monograph
. This approach is applicable to a general nonlinear nth order ordinary or partial diﬀerential
equation in N-dimensional space, and to the complicated nonlinear two-point boundary condi-
tion [1, 5]. To reach convergence, the simple guess of the zeroth iteration being equal to zero or
to one of the boundary conditions, is usually enough . This approach was applied for solving
nonlinear Calogero equation in a variable phase approach to quantum mechanics. The results
were compared with those of perturbation theory and the exact solutions. In this method iter-
ations yield rapid quadratic convergence and often monotonicity. In  the Quasilinearization
method was applied to solve diﬀerent nonlinear ordinary diﬀerential equations in physics, such
as Lane-Emden, Thomas-Fermi, Duﬃng and Blasius equations. They have shown this method
gives excellent results. These equations contain not only quadratic nonlinear terms but also
various other forms of nonlinearity of the higher derivatives. It was shown that a small number
of the QLM iterations yield fast convergence and uniformly stable numerical results.
The Volterra’s model for population growth of a species within a closed system is given in
[15, 12, 14] as
p(x)dx, p(0) = p0,(1)
1Department of Computer Science , Shahid Beheshti University, Tehran, Iran
2Department of Applied Mathematics, Shahid Beheshti University, Tehran, Iran
3Department of Medical physics and Biomedical Engineering, Shiraz University of Medical Sciences,
Manuscript received 20 September 2008.
2 APPL. COMPUT. MATH., V.9, N.1, 2010
where a > 0 is the birth rate coeﬃcient, b > 0 is the crowding coeﬃcient, and c > 0 is the
toxicity coeﬃcient. The coeﬃcient cindicates the essential behavior of the population evolution
before its level falls to zero in the long term. p0is the initial population, and p=p(˜
the population at time ˜
t. This model is a ﬁrst-order integro-ordinary diﬀerential equation where
the term cp
p(x)dx represents the eﬀect of toxin accumulation on the species. we apply scale
time and population by introducing the nondimensional variables
b, u =pb
to produce the nondimentional problem
u(x)dx, u(0) = u0,(3)
where u(t) is the scaled population of identical individuals at time t, and κ=c/(ab) is a
prescribed nondimensional parameter. The only equilibrium solution of (3) is the trivial solution
u(t) = 0 and the analytical solution 
u(t) = u0exp(1
shows that u(t)>0 for all tif u0>0. In , the singular perturbation method for Volterra’s
population model is considered. It is shown in  that for the case κ¿1, where populations are
weakly sensitive to toxins, a rapid rise occurs along the logistic curve that will reach a peak and
then is followed by a slow exponential decay. And, for κlarge, where populations are strongly
sensitive to toxins, the solutions are proportional to sech2(t).In , the series solution method
and the decomposition method are implemented independently to (3) and to a related nonlinear
ordinary diﬀerential equation. Furthermore, the Pade approximations are used in the analysis
to capture the essential behavior of the populations u(t) of identical individuals. He compared
the approximation of umax and exact value of umax for diﬀerent κ.
The solution of (1) has been of considerable concern. Although a closed form solution has
been achieved in [14, 13], but it was formally shown that the closed form solution cannot lead to
any insight into the behavior of the population evolution . Some approximate and numerical
solutions for Volterra’s population model have been reported. In reference , the successive
approximations method was oﬀered for the solution of (3), but was not implemented. In this case
the solution u(t) has a smaller amplitude compared to the amplitude of u(t) for the case κ¿1.
In  , several numerical algorithms namely the Euler method, the modiﬁed Euler method, the
classical fourth-order Runge-Kutta method and Runge-Kutta-Fehlberg method for the solution
of (3) are obtained. Moreover,a phase-plane analysis is implemented. In , the numerical
results are correlated to give insight on the problem and its solution without using perturbation
techniques. However, the performance of the traditional numerical techniques is well known
in that it provides grid points only, and in addition, it requires large amounts of calculations.
In [9, 10] applied spectral method to solve Volterra’s population on a semi-inﬁnite interval.
This approach is based on a Rational tau method. They obtained the operational matrices of
derivative and product of rational functions and reduced the solution of this problem to the
solution of algebraic equations systems.
As one more step in this direction, We use Quasilinearization approach for solving Volterra’s
population model and then we apply Baricentric Lagrange Interpolation with xj’s chebyshev
K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...3
points of the ﬁrst kind. on the other hand, comparison of results obtained by this method and
others shows that this method provides more accurate and numerically stable solution than
those obtained by the other methods.
The paper is arranged as follows: in section 2 we describe the Quasilinearization method. In
this section the Volterra’s population model is considered. This equation is ﬁrst converted to
an equivalent nonlinear ordinary diﬀerential equation. Then we apply our method to solve this
ODE. In this section, the proposed method is used for the numerical examples, and a comparison
is made with the existing methods in the literature. Then, the results and numerical tables are
stated. The Finding of the study are summarized in the last section.
2. The Quasilinearization method (QLM)
The Quasilinearization method (QLM) is an application of the Newton-Raphson-Kantrovich
approximation method in function space . This method is applied to solve a nonlinear nth
order ordinary or partial diﬀerential equation in N-dimensions as a limit of a sequence of linear
diﬀerential equations. The idea and advantage of the method is based on the fact that linear
equations can often be solved analytically or numerically using superposition principle while
there are no useful techniques for obtaining the general solution of a nonlinear equation in terms
of a ﬁnite set of particular solutions. As we discussed in the introduction, we limit ourselves here
to a nonlinear ordinary diﬀerential equation in one variable on the interval [0,b] which could be
L(n)y(x) = f(y(x), y(1)(x), ...., y(n−1)(x), x),(4)
with n boundary conditions
hi(y(0), y(1)(0), ..., y(n−1) (0)) = 0, i = 1, .., d (5)
hi(y(b), y(1)(b), ..., y(n−1)(b)) = 0, i =d+ 1, .., n, (6)
here L(n)is the linear nth order ordinary diﬀerential operator, fand h1, h2, ..., hnare the non-
linear functions of y(x) and its (n-1) derivatives are y(j)(x)j= 1, ..., n −1. The QLM
prescription [1, 5, 7] determines the (r+ 1)th iterative approximation yr+1(x) to the solution of
(4) as a solution of the linear diﬀerential equation:
L(n)yr+1(x) = f(yr(x), y(1)
r(x), ...., y(n−1)
r(x), x)+ (7)
r(x), ...., y(n−1)
r(x) = yr(x), with linearized two-point boundary conditions
r(0), ...., y(n−1)
r(0),0) = 0, k = 1, .., d (8)
r(b), ...., y(n−1)
r(b), b) = 0, k =d+ 1, .., n. (9)
Here the functions fy(j)=∂f
∂yjare functional derivatives of the functionals
f(y(x), y(1)(x), ...., y(n−1)(x), x)
4 APPL. COMPUT. MATH., V.9, N.1, 2010
hk(y(x), y(1)(x), ...., y(n−1)(x), x)
respectively. The zeroth approximation y0(t) is chosen from mathematical or physical consid-
erations. Where δyr+1 (x) and ∆yr+1(x) are the diﬀerences between two subsequent iterations
and between the exact solution and the rth iteration, respectively:
δyr+1(x) = yr+1(x)−yr(x),
∆yr+1(x) = y(x)−yr(x).
In  proved k∆yr+1(x)k≤ kk∆yr(x)k2that showed the diﬀerence between the exact solution
and the rth iteration is decreasing quadratically and δyr+1(x) for an arbitrary l < r satisﬁed the
or for l= 0, they related the (n+ 1)th order result to the 1st iterate by
Therefore the convergence depends on the quantity q1=kky1−y0k.Where the zeroth
iteration y0(x) is chosen from physical and mathematical considerations.
3. Solving Volterra’s population model
3.1. Converting Volterra’s Population Model to a Nonlinear ODE Equation. In this
section we convert Volterra’s population model (3) to an equivalent nonlinear ordinary diﬀeren-
tial equation. Let
This leads to
y0(x) = u(x),(11)
y00(x) = u0(x).(12)
Inserting (10)-(12) into (1) yields the nonlinear diﬀerential equation
κy00 (x) = y0(x)−(y0(x))2−y(x)y0(x),(13)
with the initial conditions
y(0) = 0,
y0(0) = u0.
that are obtained by using (10) and (12) respectively.
K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...5
3.2. Solution of Volterra’s Population Model. In the previous section we converted Volterra’s
Population model to a nonlinear ODE. In this section we apply the QLM procedure to solve
this model which is reduced to setting
n+1(x) = f(yn(x), y0
n(x)) + (yn+1(x)−yn(x))fyn+ (y0
where f=y0(x)−(y0(x))2−y(x)y0(x). The (n+ 1)th iteration for volterra’s population model
has the following form:
n+1(x) = y0
n(x) + yn(x)−yn+1(x)) + y0
yn+1(0) = 0, y0
n+1(x) = 0.1,(14)
for κ= 0.02,0.04,0.1,0.2 and 0.5, where yn(x) is a previous iteration which is considered to be
a known function. We solve (14) numerically using Maple software and then apply Barycentric
Lagrange Interpolation to approximate yn+1(x) as the following [2, 3]:
where xj’s are Chebyshev points of the ﬁrst kind that are given by:
xj= cos (2j+ 1)π
2m+ 2 , j = 0, .., m.
In this case, after cancelling factors independent of jwe have:
wj= (−1)jsin (2j+ 1)π
and fj’s are the solutions of (14) in xj’s. The initial guess, satisfying the boundary conditions
at zero, was suitably made for diﬀerent values of κ.
3.3. Illustrative example. We apply the presented method in this paper to examine the math-
ematical structure of u(t). In particular, we seek to study the rapid growth along the logistic
curve that will reach a peak, then followed by the slow exponential decay where u(t)→0 as
t→ ∞. The mathematical behavior deﬁned in this way was introduced by  and justiﬁed by
 by using singular perturbation methods for the inner and outer solutions. Further, these
properties were also conﬁrmed by  upon using a phase plane analysis and  by using Pade
approximations. We apply the method presented in this paper and solved (3) where u0= 0.1
and κ= 0.02,0.04,0.1,0.2, and 0.5. In Table 1, we compared the results of QLM, method and
exact values of umax for κ= 0.02,0.04,0.1,0.2 and 0.5. In this table muis the minimum QLM
iteration number required in case that the absolute value of diﬀerences among the successive
iterations is less than (10−5).
6 APPL. COMPUT. MATH., V.9, N.1, 2010
Table 1. A comparison of the presented method(QLM) and the exact values for umax
κ muQLM Exact umax
0.02 5 0.92343412 0.92342717
0.04 4 0.87381153 0.87371998
0.1 7 0.76974153 0.76974149
0.2 3 0.65905030 0.65905038
0.5 8 0.48519041 0.48519030
In Fig.1 below is shown the results of QLM calculations for κ= 0.02,0.04,0.1,0.2 and 0.5.
0 1 2 3 4 5 6
Figure 1. The results of QLM calculations for κ= 0.02,0.04,0.1,0.2 and 0.5.
In addition, Fig.1 formally shows the rapid rise along the logistic curve followed by the slow ex-
ponential decay after reaching the maximum point.
In Table 2, the resulting values using the present method together with the results obtained
in [9, 10, 15] and the exact values are presented.
Table 2. A comparison of the method in [9, 10, 15] and the presented method with the exact values for umax.
κMethod in  Method in  Method in  QLM Exact umax
0.02 0.923327 0.923463 0.90383805 0.92343412 0.92342717
0.04 0.873605 0.873708 0.86124018 0.87381153 0.87371998
0.1 0.769623 0.769734 0.76511308 0.76974153 0.76974149
0.2 0.658872 0.659045 0.65791231 0.65905030 0.65905038
0.5 0.485076 0.485188 0.48528235 0.48519041 0.48519030
K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...7
The Quasilinearization method(QLM)is very easy to apply and this method treats the non-
linear terms as perturbation about the linear ones and is not based,unlike perturbation theories,
on the existence of some kind of small parameters.
We used QLM for solving Volterra’s Population Model that in each stage,we solved yn(x) nu-
merically and applied Barycentric Lagrange Interpolation to approximate yn+1(x) with Cheby-
shev points. This interpolation is very good and accurate.
In the end we Compared this method with the other methods and found that the answers
were very accurate, numerically stable and converged fast.
We used an illustrative example to demonstrate the application of this method, and came to
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